Some Results on the Asymptotic Normality of k-Class Estimators in the Case of Many Weak Instruments

This paper analyzes conditions under which various k-class estimators are asymptotically normal in a simultaneous equations framework with many weak instruments. In particular, our paper extends the many instruments asymptotic normality results obtained by Morimune (1983), Bekker (1994), Angrist and Krueger (1995), Donald and Newey (2001), and Stock and Yogo (2003) to the case where instrument weakness is such that the concentration parameter grows slower than the number of instruments but faster than the square root of the number of instruments. That is, we obtain asymptotic normality of estimators under situations where the problem of weak instruments is more severe than that assumed in previous papers employing a many-instruments asymptotic framework. In this case, the rate of convergence of estimators, such as LIML and B2SLS, is found to depend both on the growth rate of the concentration parameter and on the number of instruments, which differs from previous results where the rate of convergence depends either on the sample size or on the number of instruments. We also show that expressions for the asymptotic covariance matrices we derive will in general involve different components depending on whether the concentration parameter approaches infinity faster than, slower than, or at the same speed as the number of instruments. An additional finding of this paper is that, for the case where the concentration parameter grows slower than the number of instruments but faster than the square root of the number of instruments, the LIML estimator can be shown to be asymptotically more efficient than the B2SLS estimator not just for the case where the error distributions are assumed to be Gaussian but for all error distributions that lie within the elliptical family. However, for non-elliptical error distributions, the relative efficiency of LIML and B2SLS will in general depend on parameter values of the moments of the error distributions